Logic is the best guide. If you are doubting whether you ought to be doing it, don't.

1) "Common sense" is not the same as logic. There are laws of logic that are the same for everyone, and "common sense" is highly subjective.

2) "If you are doubting whether you ought to be doing it, don't." - This assumes that our instincts are (always? frequently?) accurate when we know they are not.

I am not sure what this was referring to exactly (restrictions on behavior regarding COVID-19?) but depending on the behavior, some are more instinctual than others. I can see if "logic" here is being used in the colloquial sense (which could be more similar to common sense), then we can ignore my comment #1.

I think you're reading it as though it's written like this:

"This is all driven by common sense. Logic is the best guide; if you are doubting whether you ought to be doing it, don't.'

The part after the semi colon expands on the part before it. However, written as it was I think that it's just three sentences supporting the speaker's point, but which aren't necessarily predicated on each other.

Of course it depends on who wrote it and whether they conveyed what the speaker actually meant. 

Doubt simply means uncertainty of, and thus hesitation in, action or belief. It's not an indicator of whether you are right or wrong, or thinking illogically. It depends on what your doubt is based on - if you're an ardent Marxist, for example, but read from non-Marxist (or even Marx-critical) sources of information, you may start doubting - reasonably - what you read before. On the other hand, falling for an Ad Fidentia ("but are you SURE you believe that?") is unreasonable, and therefore illogical, doubt.

Common sense =/= logic mind you (Appeal to Common Sense is actually fallacious since things that are intuitive to humans, or 'common sense', may turn out to be horridly false; so it serves as a thought-terminating cliché).

Doubt is defined as a feeling of uncertainty. So, It seems like this would be a kind of an Appeal To Emotion/feeling Fallacy. 

Logical Form:

X doest feel right.

Therefore X is not right. 

Feelings have no barring on whether something is right or wrong, or true or false.